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Scenario Building (6) Scenario Building (Specific Examples and Interpretation)– For example, a scenario assuming that “the presumed amount of loss comesto 1 billion and this loss is likely to occur once in every twenty years” isestablished in a cell (a combination of a business line and event type): This scenario is quite different from the assumption that “the presumed amount ofloss is 50 million and this loss is likely to occur once a year or so.” Supposing that the average number of accidents per year in this cell is 50 cases, itis interpreted that approximately 1,000 accidents are expected to occur in twentyyears’ time and one of the accidents (the worst case) causes 1 billion in loss.– A natural interpretation is that the probability of the amount of loss from theaccident exceeding 1 billion is 0.1% or so.– Conversely speaking, does it mean that the 99.9% VaR for the cell is literallyset at 1 billion? An ideal approach should allow giving the anticipated frequency of “once in everyso-and-so cases processed” or “once in every so-and-so working hours” from thebottom-up analysis, rather than “once in so-and-so years” .– A possible approach is to analyze the accident occurrence pattern from theperspective of the study of failure.– Is overwork or inattention the cause of error?– Is a newly installed system unstable?– What about a planned change in the business style?2006/7/1816

Scenario Building (7) Scenario Building (How to Use Data)– How to use scenario data: A risk audit system that allows you to build a good scenario may well bean important asset for risk management. An extreme case should be considered through stress testing byincluding correlativity. A distribution of losses per case is estimated in combination with internaldata.– It is doubtful whether the two factors could be used on an equal footing. A proposed method is (although it is not statistically recognized):– First of all, internal data are arranged in ascending order of the amount ofloss and a probability based on the corresponding cumulative experiencedistribution is given to each loss data.– Since scenario data are based on the assumption that the anticipatedprobability corresponding to the amount of loss is given, the pair of theamount of loss and the anticipated probability are fused with internal data.– Based on the fused data set, a distribution of losses is estimated. However,attention should be given to the possibility of difference between the scenariodata and the estimates.2006/7/1817

An Example of the Scenario Utilization Method UsingExtreme Value Theory (1) 2006/7/18When VaR is regarded as a measure of risks, a major influencemay come not from an accident involving low losses but from a“huge” loss accident, if it occurs infrequently.In this case, it may be reasonable to focus critically on the lowerright part of the distribution, rather than to build a refined modelfor the whole distribution of losses.Application of Extreme Value Theory (EVT)This is a common theme in papers discussing the metrization ofoperational risks from the perspective of a statistical approach.A model that applies a method called the Peak-Over-Threshold(POT) approach is described here.18

An Example of the Scenario Utilization Method UsingExtreme Value Theory (2)Whole DataA Schematic Description of the POT Approach12000010000080000Only data in excess of thethreshold set at 10,000 relativeto a certain data set arehandled.6000040000200000An excess above the thresholdof 10,000 is shown.Excess Data Set (Threshold 10,000)12000010000080000600004000020000An empirical distribution ofexcesses is prepared byarranging excesses inascending order and assigningappropriate probability to theseexcesses ( in the figure).0Excess Above Threshold and Empirical 6/7/1880000GPD(0.25,13000)100000120000140000This empirical distribution ofexcesses is approximated bythe Generalized ParetoDistribution (GPD).19

An Example of the Scenario Utilization Method UsingExtreme Value Theory (3) Distribution function in the Generalized Pareto Distribution: ξ 11 β G ( x; ξ , β ) 1 exp ξ1x (ξ 0) x (ξ 0)β x 0 (ξ 0) , β 01 x0(ξ0) ξ – Asξ increases, the base becomes thicker (even the average doesnot exist when ξ １). Excess distribution function (relative to u):X : As a probability variable representing the amount of loss from the accident :P( X x u ) P( X u )P( X u x X u ) P( X u )2006/7/1820

An Example of the Scenario Utilization Method UsingExtreme Value Theory (4) Theoretically no matter what original loss distributionmay be, the excess distribution function relative to asufficiently high threshold u can be approximated byGPD (as long as certain conditions are satisfied).P ( X u x X u ) G ( x; ξ , β )for x u– However, a threshold must be determined carefully becausethe theory does not specify the degree of u.– For the sake of accuracy, it is desirable to verify the validityof the empirical distribution of high loss data and GPD bycarrying out goodness-of-fit tests such as KolmogorovSmirnov and Anderson-Darling tests.2006/7/1821

An Example of the Scenario Utilization Method UsingExtreme Value Theory (5) What to do with a method of estimating parameters for GPD ・・・– Maximum Likelihood Method: If numerous samples are available, an estimate will have a theoreticallydesirable nature but what if small samples and identical numbers areobtained in clusters? When estimating parameters using a scenario, it is difficult to simplyapply this method if scenario data are treated similarly to internal data.– Moments Method: This method is often used to estimate parameters when few data areavailable. This method cannot apply to some distributions in which moments ofhigher degree do not exist (e.g., when ξ １ in GPD). When estimating parameters using a scenario, it is difficult to simplyapply this method if scenario data are treated similarly to internal data.– Least Square Method (calibration rather than estimation): Given an equal weight, entire goodness of fit increases but there mayrather be a larger discrepancy to occur at the low-frequency and highloss portion that is important in risk management. If a scenario is used, this method distinguishes it from internal data bythe size of “an empirical cumulative probability.”2006/7/1822

An Example of the Scenario Utilization Method UsingExtreme Value Theory (6) Probability-Weighted Moments Method– This is a method of obtaining a parameter as a solution to an equation, aswith the Moments Method, by determining an expected value that isweighted by the exponentiation of the probability of a value exceeding avalue calculated from the presumed probability distribution (this probabilityshould be very low) and comparing it with a value derived similarly fromactual data.– This method is applicable when moments of higher degree do not exist. Itis reportedly empirically effective when small samples are given. However, it is difficult to apply this method unless the followingexpected value can actually be calculated:E[ Z (1 F ( Z ;θ )) r ] ( Z： Probability variable, F： Distribution function that Z follows)– The way “an empirical cumulative probability” is determined has aninfluence on the results (arbitrariness is involved): It is related to an assessment of the frequency rate of external data. The results of estimation with a scenario incorporated may likely bedifferent from the original scenario.2006/7/1823

Conclusion (1) An optimum combination of a distribution and anestimation method cannot be presented in generalterms:– Supporting with a statistical theory is difficult.– There are no absolute criteria. Is it good to remain at the same level as it was or should itbe overestimated? Difficult estimation of a distribution of losses usingexternal cases and scenarios:– Can scaling and an assessment of the incidence rate beimplemented properly?– The application of a huge loss case may have a very largeimpact on the estimation process and result in leaving amodel out of consideration.2006/7/1824

Conclusion (2) There may probably be no complete model.– Continuous discussion about an operational risk metricmodel will lead to gradual improvements in methods.– The business process should be scrutinized and, at thesame time, the forecasting of errors based on the study offailure, psychology and systems theory must be adopted.– In the final analysis, it is necessary to fully analyze the dataon accidents both qualitatively and quantitatively. Theremust be a system under which wisdom is shared to addressthe issue not as a problem for individual financial institutionsbut as a problem for the financial industry as a whole.2006/7/1825

REFERENCES 2006/7/18Embrechts, P., Klüppelberg, C., Mikosch, T., “Modelling ExtremeEvents”, Springer (1997)Embrechts, P., Furrer, H., Kaufmann, R., “Quantifying RegulatoryCapital for Operational Risk. Derivatives Use”, Trading & Regulation9(3), 217-233 (2003)Frachot, A., Georges, P., Roncalli, T., “Loss Distribution Approach forOpérational Risk”, Working Paper, Groupe de RechercheOpérationnelle, Crédit Lyonnais (2001)King, J. L., “Operational Risk: Measurement and Modelling” , JohnWiley & Sons (2001)– “Operational Risk Management,” translated by Saito, Haruhiko,Oguro, Naoki, Sigma Bay’s Capital (2002)Operational Risk Research Institute, Mitsubishi Trust and BankingCorporation, “The Practice of Operational Risk Management,” ToyoKeizai Shinposha (2002)26

Reference: Poisson Distribution Probability of an event taking place k times:P (k ) e λ λkλ 0,k!The Poisson distribution is used as a demonstrative frequency ratemodel when the probability of an event is low and the event occurs dueto an independent factor or factors.The average and dispersion parameters both are expressed by theintensity parameter λ.Possible extended applications are to make the intensity parametertime-dependent such as λ(t) and to use the estimates of external data.– Seasonality, the startup of a new business, and the expansion of businessoperations are taken into /18123456789 10 11 12 13 14 1527

Reference: Logarithmic Normal Distribution (log x µ ) 2 f ( x; µ , σ ) exp 22σ2π σ x log x µ F ( x; α , β ) N x 0, σ 0 σ 1 This distribution is used, for example, to represent a distribution offuture stock prices in financial engineering.It is easy to handle for a model that has a distribution with a thick tail.LN(1,2)0.60.50.40.30.20.1002006/7/18123428

Reference: Weibull Distribution x α α 1f ( x;α , β ) α xexp β β α xF ( x ; α , β ) 1 exp β α x 0, α 0, β 0It is used for survival analysis, for example.The density function shows a tendency that the greater α is, the steeper theslope becomes, and the smaller β is, the greater the degree of 0.05002006/7/1812345629

Reference: Gamma Distributionf ( x;α , β ) Γ( x ) 0 x α 1xexp αβ Γ (α ) β 1x 0, α 0, β 0e u u x 1duIt is used for survival analysis, for example.The density function shows a tendency that the greater α is, the gentler theslope becomes and the thicker the tail 45630

Reference: POT Approach (1)The distributi on function for the whole is given as F(x).(Although F(x) must meet some conditions , a well - knownfunction poses almost no problem. ) A threshol d is given as uand the excess distributi on function for H relative to u isconsidered :F ( x u ) F (u )Fu ( x ) 1 F (u )Then, when Fu (x) is u x F : a theoretic al end,it is known that a certain ξ, β can be uniformly approximat edby GPD : ξG ( x; ξ , β ) 1 1 β2006/7/18 ξ1x (u x x F )31

Reference: POT Approach (2)Here, for x u, the distributi on function can be expressed bythe approximat e expression :F ( x ) (1 F (u ))G ( x u; ξ , β ) F (u )When the number of data for the whole is given as N andthe number of data exceeding the threshold u is set as n(u),n(u)then F(u) can be approximat ed by 1 NUsing this, the function is approximat ed as :n (u ) x u F ( x) 1 1 ξN β 2006/7/18 1ξ( x u)32

Reference: POT Approach (3) An Estimation Approach Using the Probability-WeightedMoments (PWM) Method Relative to GPDF(x; ξ, β) is given as the distribution function for GPD.Z is the probability variable that follows this distributionfunction. Thus it should be noted that for the natural numberr(the number satisfying r 1 ξ 0) the following equationholds true :βE[ Z (1 F ( Z ; ξ , β )) r ] (r 1 ξ )(r 1)On the other hand, the data x1 , L , x N (arranged in ascending order)is assumed to follow GPD. Where,1wˆ r N2006/7/18N x (1 F ( x ; ξ , β ))nrnn 133

Reference: POT Approach (4)Here when the uniform random number sequence U1 L , U N (arrangedin ascending order) on (0, 1) is taken, it is known that the following nature holds true(Quantile transformation lemma) :(d )( F ( x1 ; ξ , β ),L , F ( x N ; ξ , β )) (U1 L , U N )Therefore, it can be considered as :1 Nwˆ r xn (1 U n ) rN n 1Actually, instead of the uniform random number sequence,the plotting points :pn N n δnN n 0.5or generally pn NN γ nare given as appropriate values.2006/7/1834

Reference: POT Approach (5)Finally,1 Nwˆ r xn (1 pn ) rN n 1is regarded as " the probability weighted moment of the sample" ,a theoretical value and an equation are generated, and a solutionto this equation is given as an estimate.In the present analysis, ξ 1 is assumed (namely theaverage exists), and with r 0,1, an equation is created. An estimatecan be obtained as :2 wˆ 0 wˆ 1wˆ 0, β 2 ξ 2 wˆ 0 2 wˆ 1wˆ 0 2 wˆ 12006/7/1835

- Combination with scenario analysis may not make it easy to apply a parametric method. - However, a method of combining scenarios (including scaled scenario of external cases) with internal data and calculating the operational risk 99.9% VaR with a parametric technique may not be totally impossible. Practical Problems (3)